Fast 3D forward modeling method and system with second-order calculation accuracy for non-uniform grid models

Abstract

The present invention belongs to the field of geophysical exploration, and provides a fast three-dimensional forward modeling method and system with second-order calculation accuracy of a non-uniform grid model. The technical scheme is as follows: determine the non-uniform grid discretization mode of the three-dimensional forward numerical model calculation area, and divide the three-dimensional forward numerical model calculation area; based on the divided calculation area, use the central difference method of non-adjacent field values ​​to calculate the magnetic field partial derivative at the edge of the non-uniform grid, so that the electromagnetic field calculation result at this position has second-order calculation accuracy; assemble the calculation equations of all discrete electric and magnetic fields in the calculation area to obtain the electromagnetic field full discrete equation with second-order calculation accuracy; solve the equation to obtain the electric and magnetic field values ​​at the sampling point in the measurement space. The calculation accuracy is improved without reducing the calculation efficiency.

Description

Technical Field

[0001] The invention belongs to the field of geophysical exploration, and in particular relates to a fast three-dimensional forward modeling method and system for second-order calculation accuracy of a non-uniform grid model. Background Art

[0002] The statements in this section merely provide background information related to the present invention and do not necessarily constitute prior art.

[0003] The electromagnetic field of the transient electromagnetic method is a diffusion field generated by an excitation source and diffused to infinity. When using a computer to simulate the electromagnetic field in this method, it is necessary to consider using a limited calculation area to simulate the electromagnetic diffusion phenomenon in real space. Therefore, an artificial boundary condition (Dirichlet boundary condition) is introduced to force the boundary of the numerical model to be truncated, that is, the tangential component of the electric field at the boundary of the numerical model is 0. However, since the emission waveform used in the transient electromagnetic method is a wide-band trapezoidal wave, the electromagnetic field containing low-frequency components requires a long distance in the lossy medium to consume the electromagnetic field generated by the emission source, so the numerical model requires a larger calculation area.

[0004] When using structured grid to discretize the calculation area, it can be divided into uniform grid and non-uniform grid division according to the size of the model unit. Uniform grid division means that the grid size of the entire calculation area is equal. In the uniform grid division model, the traditional central difference method is used to discretize the curl operator in the Maxwell equations, and the calculation results obtained have second-order calculation accuracy. However, using uniform grid division to divide the calculation area will result in a large number of grids, thereby reducing the forward calculation speed.

[0005] Therefore, non-uniform meshing strategies are often used for large-scale calculation areas. This method uses a smaller uniform mesh in the core calculation area (including the emission source and observation point location). Outside the core calculation area, the mesh size gradually increases according to a proportional coefficient greater than 1 until the model boundary. The non-uniform meshing strategy can use a smaller number of grids to fill the calculation area, thereby significantly improving the forward calculation speed. However, when the traditional central difference method is used to calculate the partial derivatives of the magnetic field at the boundary of the non-uniform grid, the second-order error term of the grid size of the calculation formula of the electromagnetic field partial derivative obtained by Taylor expansion is not 0, so the calculation result only has the first-order accuracy of the grid size, and the calculation accuracy is low. Summary of the invention

[0006] In order to solve at least one technical problem existing in the above-mentioned background technology, the present invention provides a fast three-dimensional forward modeling method and system with second-order calculation accuracy of a non-uniform grid model, which uses non-adjacent electromagnetic fields to calculate the partial derivatives of the electromagnetic field at the edges of the non-uniform grid, so that the electromagnetic field calculation results at this position have second-order calculation accuracy. This method improves the calculation accuracy without reducing the calculation efficiency.

[0007] In order to achieve the above object, the present invention adopts the following technical solution:

[0008] A first aspect of the present invention provides a fast three-dimensional forward modeling method with second-order calculation accuracy for a non-uniform grid model, comprising the following steps:

[0009] Determine the non-uniform grid discretization method of the three-dimensional forward numerical model calculation area, and divide the three-dimensional forward numerical model calculation area;

[0010] Based on the divided calculation area, the central difference method of non-adjacent field values ​​is used to calculate the partial derivatives of the magnetic field at the edges of the non-uniform grid, so that the electromagnetic field calculation results at this location have second-order calculation accuracy;

[0011] The calculation equations of all discrete electric and magnetic fields in the calculation area are assembled to obtain the full discrete equation of the electromagnetic field with second-order calculation accuracy; the electric and magnetic field values ​​at the sampling points in the calculation space are obtained by solving the equation.

[0012] Furthermore, the non-uniform grid discretization method of the three-dimensional forward numerical model calculation area is: assuming that the model is x The direction is uniformly meshed, and the mesh size is ,exist y The direction is non-uniform grid division, and the non-uniform grid interface y = y 1 The left side of the grid is The grid size is of the grid, and .

[0013] Furthermore, the central difference method of non-adjacent field values ​​is used to calculate the partial derivatives of the magnetic field at the edges of the non-uniform grid, so that the electromagnetic field calculation results at this position have second-order calculation accuracy, including:

[0014] Determine the size and The positional relationship of the grid;

[0015] Calculate the Taylor expansion of the discrete magnetic field at the center of the left grid surface at the non-uniform grid interface;

[0016] based on and The relationship between them is used to calculate the Taylor expansion of the discrete magnetic field at the center of the non-adjacent grid surface on the right side of the non-uniform grid interface;

[0017] The partial differential operator of the magnetic field at the grid edge is obtained by subtracting the Taylor expansion of the discrete magnetic field at the center of the non-adjacent grid surface on the right side of the non-uniform grid interface from the Taylor expansion of the discrete magnetic field at the center of the grid surface on the left side.

[0018] Furthermore, the non-uniform mesh size that satisfies the second-order accuracy of the calculation results of the partial derivatives of the magnetic field at the edges of the non-uniform mesh is and The relationship between them is: , is an integer.

[0019] Furthermore, the partial derivative of the magnetic field at the edge of the non-uniform grid is:

[0020] ,

[0021] In the formula, is the partial derivative of the magnetic field at the edge of the non-uniform grid, represents the discrete magnetic field at the center of the left grid surface, represents the discrete magnetic field at the center of the right grid surface, Indicates that the magnetic field y 1 The second-order derivative at , represents the higher-order error term with respect to the mesh size.

[0022] Furthermore, the fully discrete equation of the electromagnetic field with second-order calculation accuracy is:

[0023] ,

[0024] ,

[0025] ,

[0026] ,

[0027] ,

[0028] ,

[0029] In the formula, i , j and k is the index number of the discrete electromagnetic field in space, where The index number representing the position of the non-uniform grid interface. and They are x and z Directional grid size, express The ratio of expressx The virtual electric field in the direction of express y The virtual electric field in the direction of express z The virtual electric field in the direction of express n time x The electric field in the direction of express n time y The electric field in the direction of express n time z The electric field in the direction of express n time x Direction of the magnetic field, express n time y Direction of the magnetic field, express n time z Direction of the magnetic field, express n+ 1 moment x The electric field in the direction of express n+ 1 moment y The electric field in the direction of express n+ 1 moment z The electric field in the direction y The direction is non-uniform grid division, and the non-uniform grid interface y = y 1 The left side of the grid is The grid size is The grid, , , , , and are the vacuum magnetic permeability and vacuum dielectric constant, respectively. σ is the conductivity, is the step time.

[0030] A second aspect of the present invention provides a fast three-dimensional forward modeling system with second-order computational accuracy for a non-uniform grid model, comprising:

[0031] A calculation region partitioning module is used to determine a non-uniform grid discretization method for the calculation region of the three-dimensional forward numerical model and partition the calculation region of the three-dimensional forward numerical model;

[0032] The second-order calculation accuracy module is used to calculate the partial derivatives of the magnetic field at the edges of the non-uniform grid based on the divided calculation area using the central difference method of non-adjacent field values, so that the electromagnetic field calculation results at this position have second-order calculation accuracy;

[0033] The three-dimensional forward modeling module is used to discretize the constructed Maxwell equations in the space domain and time domain using the central difference method and the backward Euler method respectively, to obtain the fully discrete equations of the BEDS-FDTD forward algorithm, and combine the partial derivatives of the electromagnetic field at the edges of the non-uniform grid with second-order calculation accuracy to obtain the fully discrete equations of the electromagnetic field with second-order calculation accuracy; the electric and magnetic field values ​​at the sampling points in the measurement space are calculated based on the fully discrete equations of the electromagnetic field with second-order calculation accuracy.

[0034] Furthermore, in the calculation area partitioning module, the non-uniform grid discretization method of the calculation area of ​​the three-dimensional forward numerical model is: assuming that the model is x The direction is uniformly meshed, and the mesh size is ,exist y The direction is non-uniform grid division, and the non-uniform grid interface y = y 1 The left side of the grid is The grid size is of the grid, and .

[0035] Furthermore, in the second-order calculation accuracy module, the central difference method of non-adjacent field values ​​is used to calculate the partial derivatives of the magnetic field at the edges of the non-uniform grid, so that the electromagnetic field calculation results at this position have second-order calculation accuracy, including:

[0036] Determine the size and The positional relationship of the grid;

[0037] Calculate the Taylor expansion of the discrete magnetic field at the center of the left grid surface at the non-uniform grid interface;

[0038] based on and The relationship between them is used to calculate the Taylor expansion of the discrete magnetic field at the center of the non-adjacent grid surface on the right side of the non-uniform grid interface;

[0039] The partial differential operator of the magnetic field at the grid edge is obtained by subtracting the Taylor expansion of the discrete magnetic field at the center of the non-adjacent grid surface on the right side of the non-uniform grid interface from the Taylor expansion of the discrete magnetic field at the center of the grid surface on the left side.

[0040] Furthermore, in the second-order calculation accuracy module, the non-uniform mesh size that satisfies the second-order accuracy of the calculation results of the partial derivatives of the magnetic field at the edges of the non-uniform mesh is and The relationship between them is: , is an integer.

[0041] Compared with the prior art, the present invention has the following beneficial effects:

[0042] 1. The present invention adopts the BEDS-FDTD fast three-dimensional forward algorithm for calculating the electromagnetic field at the edge using non-adjacent field values. The algorithm uses non-adjacent field values ​​to discretize the partial differential operators of the electromagnetic field in the spatial domain at the boundary of the non-uniform grid, and uses adjacent field values ​​to discretize the partial differential operators of the electromagnetic field in the spatial domain at the boundary of the uniform grid. Through this differential discretization method, the partial differential operators in the spatial domain can be differentially discretized to maintain second-order calculation accuracy.

[0043] 2. Application of three-dimensional forward modeling algorithm for calculating electromagnetic fields at edges of non-adjacent field values: The present invention can improve the calculation accuracy of the BEDS-FDTD algorithm in non-uniform grids. When the calculation area is divided into non-uniform parts, the algorithm improves the calculation accuracy of the forward modeling results without reducing the calculation efficiency.

[0044] Advantages of additional aspects of the present invention will be given in part in the following description, and in part will become obvious from the following description, or will be learned through practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] The accompanying drawings in the specification, which constitute a part of the present invention, are used to provide a further understanding of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute improper limitations on the present invention.

[0046] Figure 1 It is a flow chart of a fast three-dimensional forward modeling method of second-order calculation accuracy of a non-uniform grid model provided by an embodiment of the present invention;

[0047] Figure 2 Schematic diagram of non-uniform grid division provided by an embodiment of the present invention, wherein (a) is a schematic diagram of calculating the partial derivative of the magnetic field on the boundary of a non-uniform grid using a traditional central difference method, and (b) is a schematic diagram of the partial derivative of the magnetic field at the interface of discrete magnetic field calculation of non-adjacent grids when the grid expansion ratio is 3;

[0048] Figure 3 is a uniform half-space model provided by an embodiment of the present invention;

[0049] Figure 4 The embodiment of the present invention provides b z / d tComparison of calculation results and relative errors, where (a) is the electromagnetic response characteristics calculated by different algorithms, and (b) is the relative error comparison between the calculation results of the traditional BEDS-FDTD algorithm and the modified BEDS-FDTD algorithm and the uniform grid calculation results. DETAILED DESCRIPTION

[0050] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0051] It should be noted that the following detailed descriptions are all illustrative and intended to provide further explanation of the present invention. Unless otherwise specified, all technical and scientific terms used herein have the same meanings as those commonly understood by those skilled in the art to which the present invention belongs.

[0052] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit exemplary embodiments according to the present invention. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprising" and / or "including" are used in this specification, it indicates the presence of features, steps, operations, devices, components and / or combinations thereof.

[0053] When a non-uniformly sized structured grid is used to divide the calculation area, the traditional backward Euler direct splitting finite difference time domain (BEDS-FDTD) method cannot obtain a second-order accuracy calculation result. The present invention aims to improve the numerical accuracy of the calculation results of the non-uniform grid division model. The present invention adopts the BEDS-FDTD fast three-dimensional forward algorithm for calculating the electromagnetic field at the edge using non-adjacent field values, and uses non-adjacent field values ​​to discretize the electromagnetic field partial differential operator in the space domain at the boundary of the non-uniform grid, and uses adjacent field values ​​to discretize the electromagnetic field partial differential operator in the space domain at the boundary of the uniform grid. The differential discretization method can make the partial differential operator in the space domain differentially discretized to maintain the second-order calculation accuracy. The present invention can improve the calculation accuracy of the BEDS-FDTD algorithm in a non-uniform grid. When a non-uniformly divided calculation area is used, the algorithm improves the calculation accuracy of the forward modeling results without reducing the calculation efficiency.

[0054] Embodiment 1

[0055] like Figure 1 As shown, this embodiment provides a fast three-dimensional forward modeling method with second-order calculation accuracy for a non-uniform grid model, comprising the following steps:

[0056] Step 1: Determine the non-uniform grid discretization method of the three-dimensional forward numerical model calculation area and divide the three-dimensional forward numerical model calculation area;

[0057] The non-uniform grid discretization method of the calculation area of ​​the three-dimensional forward numerical model is as follows: Figure 2 As shown, Figure 2 The triangles in the middle represent discrete electric fields, and the circles represent discrete magnetic fields. (a) is a schematic diagram of the calculation of partial derivatives of the magnetic field on the boundary of a non-uniform grid using the traditional central difference method, and (b) is a schematic diagram of the calculation of partial derivatives of the magnetic field at the interface of discrete magnetic field calculations of non-adjacent grids when the grid expansion ratio is 3.

[0058] Assume that the calculation area of ​​the three-dimensional forward numerical model is x The direction is uniformly meshed, and the mesh size is .exist y The direction is non-uniform grid division, and the non-uniform grid interface y = y 1 The left side of is a larger grid, the grid size is ; The right side of the interface is a smaller grid, the grid size is .

[0059] Step 2: Based on the spatial position of the discrete electromagnetic field after the calculation area is discretized, Taylor expansion is performed on the discrete magnetic field at the grid centers on both sides of the non-uniform grid interface;

[0060] In this embodiment, when the discrete magnetic field at the grid center on both sides of the non-uniform grid interface is used for calculation (the magnetic field at the position of the red circle), the discrete magnetic field at the grid center on both sides of the non-uniform grid interface is Taylor expanded, and the Taylor expansion expressions are respectively:

[0061] (1),

[0062] (2),

[0063] In the formula, represents the discrete magnetic field at the center of the right grid surface, Represents the discrete magnetic field at the center of the left grid surface; Represents a non-uniform mesh interface y 1 The discrete magnetic field at and Respectively represent the magnetic field in y 1 The first and second derivatives at ; represents the higher-order error term with respect to the mesh size.

[0064] Step 3: Based on the divided calculation area, the central difference method of non-adjacent field values ​​is used to calculate the partial derivatives of the magnetic field at the edges of the non-uniform grid, so that the electromagnetic field calculation results at this location have second-order calculation accuracy;

[0065] When the traditional central difference method is used to calculate the partial derivative of the magnetic field at the edge of a non-uniform grid, the difference between the discrete magnetic field at the center of the right grid surface and the discrete magnetic field at the center of the left grid surface is used. That is, the expression of the partial derivative of the magnetic field at the edge of the non-uniform grid is obtained by subtracting formula (2) from formula (1):

[0066] (3),

[0067] According to formula (3), the second term on the right side of the equation is When it is 0, the calculation result of this formula has second-order calculation accuracy, but observation Figure 1 From (a), we can see , so the calculation result has only first-order accuracy.

[0068] Larger grid size With smaller grid size The relationship between them is: , is an integer;

[0069] Partial derivatives of magnetic field at the edges of non-uniform grids for:

[0070] ,

[0071] For example, when the larger grid size With smaller grid size The relationship between them is: ,observe Figure 1 The Taylor expansion expression of the magnetic field at the red circle in the center of the right grid surface in (b) is:

[0072] (4),

[0073] Subtracting formula (2) from formula (4) yields the partial derivative of the magnetic field at the edge of the non-uniform grid: :

[0074] (5),

[0075] Observe the second term on the right side of equation (5), which is 0. Therefore, the result of this equation is second-order accurate. This equation will be used to calculate the first-order partial derivative of the magnetic field and the second-order partial derivative of the electric field at the boundary of the non-uniform grid.

[0076] The present invention uses non-adjacent electromagnetic fields to calculate the partial derivatives of the electromagnetic field at the edges of the non-uniform grid, so that the electromagnetic field calculation results at this position have second-order calculation accuracy. This method improves the calculation accuracy while not reducing the calculation efficiency.

[0077] Step 4: Assemble the calculation equations of all discrete electric and magnetic fields in the calculation area to obtain the full discrete equation of the electromagnetic field with second-order calculation accuracy;

[0078] The specific steps include:

[0079] Step 401, constructing the Maxwell equations;

[0080] Maxwell's equations include Faraday's law of electromagnetic induction and Ampere's circuit theorem:

[0081] (6),

[0082] (7),

[0083] In the formula and are the electric and magnetic fields; and are vacuum magnetic permeability and vacuum dielectric constant respectively; is the current density, is the source current density; represents the curl operator.

[0084] Step 402: The space domain and time domain of the Maxwell equations are discretized using the central difference method and the backward Euler method respectively, to obtain the fully discrete equations of the BEDS-FDTD forward algorithm:

[0085] (8),

[0086] (9),

[0087] (10),

[0088] In the formula, , , , , and are the vacuum magnetic permeability and vacuum dielectric constant, respectively. is the conductivity, is the step time; n and n +1 means n and n +1 calculation time; Indicated in Directional difference operator, For Directional difference operator, For Directional difference operator, The spatial rotation operator is expressed as x , y and z Direction, such as but = z and = y ; for The second-order central difference operator in the direction, for The second-order central difference operator in the direction, for The virtual electric field in the direction of for The virtual electric field in the direction of for n time The electric field in the direction of for n time The electric field in the direction of for n+ 1 moment The electric field in the direction of for n time Direction of the magnetic field, for n time Direction of the magnetic field, for n+ 1 moment Direction of the magnetic field, for n time Direction of the magnetic field, for n+ 1 moment Direction of the electric field.

[0089] Step 403, based on the fully discrete equation of the BEDS-FDTD forward algorithm, the electric field and magnetic field at any time at the full-space sampling point with second-order calculation accuracy can be finally obtained. The specific analysis process is as follows: Since there is only the first-order partial derivative of the electric field in formula (10), there is no need to modify this formula when the model adopts non-uniform grid partitioning;

[0090] When considering x and z The uniform grid is used in the direction. yWhen non-uniform grid division is used in the direction, formula (8) and formula (9) are x , y and z The expansion forms in three directions are:

[0091] (11),

[0092] (12),

[0093] (13),

[0094] (14),

[0095] (15),

[0096] (16),

[0097] In the formula, i , j and k is the index number of the discrete electromagnetic field in space, where j 1 The index number representing the position of the non-uniform grid interface. and They are x and z Directional grid size, express The ratio of express x The virtual electric field in the direction of express y The virtual electric field in the direction of express z The virtual electric field in the direction of express n time x The electric field in the direction of express n time y The electric field in the direction of express n time z The electric field in the direction of express n time x Direction of the magnetic field, express n time y Direction of the magnetic field, express n time z Direction of the magnetic field, express n+1 moment x The electric field in the direction of express n+ 1 moment y The electric field in the direction of express n + 1 moment z In this embodiment, =1 / 3.

[0098] Step 5: Calculate the electric field and magnetic field values ​​at the sampling points in the measurement space based on the calculated electromagnetic field full discrete equation with second-order calculation accuracy.

[0099] Through the above scheme, the present invention can improve the calculation accuracy of the BEDS-FDTD algorithm in a non-uniform grid. When a non-uniform partitioning calculation area is adopted, the algorithm improves the calculation accuracy of the forward modeling result without reducing the calculation efficiency.

[0100] Effect verification

[0101] Establish as Figure 3 The uniform half-space model shown in the figure has the following model parameters: the red box is the emission source arranged on the ground, the emission source size is 100m×100m, the emission current is 1A, and the trapezoidal wave is used as the emission waveform. During the numerical simulation, a non-uniform time step is used for calculation, and the total time step is 66037. The model size is 8420m×8420m×5520m, the 420m×420m×420m part of the center of the model is the core calculation area, and the rest is the external calculation area. First, a uniform grid is used to divide the entire calculation area, the grid size is 20m×20m×20m, the total number of grids is 421*421*276=48918516, and the calculation time is 6491.62s. When the non-uniform grid strategy is used, the core calculation area of ​​the model is divided into 20m×20m×20m uniform grids, and the three directions outside the core calculation area are divided into non-uniform grids. The ratio of adjacent grid sizes is set to 3, and the number of grids is 43*43*40=73960. The calculation time is 21.28s when the traditional central difference method is used, and the calculation time is 23.21s when the non-adjacent electromagnetic field calculation method of the electromagnetic field partial derivative at the edge of the non-uniform grid is used.

[0102] The results and relative errors of the uniform half-space model calculated using the traditional and improved BEDS-FDTD methods are shown in Figure 2. Figure 4 As shown, (a) is the electromagnetic response characteristics calculated by different algorithms, and (b) is the relative error comparison diagram between the calculation results of the traditional BEDS-FDTD algorithm and the modified BEDS-FDTD algorithm and the uniform grid calculation results.

[0103] The calculation results of the traditional BEDS-FDTD method are represented by black lines with solid symbols, and the calculation results of the improved BEDS-FDTD method are represented by black lines with hollow symbols. The black lines without any marks are the calculation results using uniform grids. The relative error of the calculation results of the traditional FDTD algorithm increases with the increase of calculation time, and the maximum relative error is about 2.85%. This is because the truncation error increases with the increase of grid size (as shown in Equation (5)). For the improved BEDS-FDTD algorithm, its maximum relative error is only about 1.09%, and in most cases, the relative error of the improved algorithm is generally smaller than the calculation results of the traditional BEDS-FDTD algorithm, especially for the late calculation results, the improved method has better calculation accuracy.

[0104] Therefore, the calculation method proposed in the present invention has higher calculation accuracy, and by comparing the calculation time, it can be seen that the modified BEDS-FDTD algorithm has a similar calculation time as the traditional BEDS-FDTD algorithm.

[0105] Embodiment 2

[0106] This embodiment provides a fast three-dimensional forward modeling system with second-order calculation accuracy for a non-uniform grid model, including:

[0107] A calculation region partitioning module is used to determine a non-uniform grid discretization method for the calculation region of the three-dimensional forward numerical model and partition the calculation region of the three-dimensional forward numerical model;

[0108] The second-order calculation accuracy module is used to calculate the partial derivatives of the magnetic field at the edges of the non-uniform grid based on the divided calculation area using the central difference method of non-adjacent field values, so that the electromagnetic field calculation results at this position have second-order calculation accuracy;

[0109] The three-dimensional forward modeling module is used to assemble the calculation equations of all discrete electric and magnetic fields in the calculation area to obtain the fully discrete electromagnetic field equation with second-order calculation accuracy; the equation is solved to obtain the electric and magnetic field values ​​at the sampling points in the measurement space.

[0110] Furthermore, in the calculation area partitioning module, the non-uniform grid discretization method of the calculation area of ​​the three-dimensional forward numerical model is: assuming that the model is x The direction is uniformly meshed, and the mesh size is ,exist The direction is non-uniform grid division, and the non-uniform grid interface The left side of the grid is The grid size is of the grid, and .

[0111] Furthermore, in the second-order calculation accuracy module, the central difference method of non-adjacent field values ​​is used to calculate the partial derivatives of the magnetic field at the edges of the non-uniform grid, so that the electromagnetic field calculation results at this position have second-order calculation accuracy, including:

[0112] Make sure the size is and The positional relationship between the non-uniform mesh grids;

[0113] Calculate the Taylor expansion of the discrete magnetic field at the center of the left grid surface at the non-uniform grid interface;

[0114] based on and The relationship between them is used to calculate the Taylor expansion of the discrete magnetic field at the center of the non-adjacent grid surface on the right side of the non-uniform grid interface;

[0115] The partial differential operator of the magnetic field at the grid edge is obtained by subtracting the Taylor expansion of the discrete magnetic field at the center of the non-adjacent grid surface on the right side of the non-uniform grid interface from the Taylor expansion of the discrete magnetic field at the center of the grid surface on the left side.

[0116] Furthermore, in the second-order calculation accuracy module, the non-uniform mesh size that satisfies the second-order accuracy of the calculation results of the partial derivatives of the magnetic field at the edges of the non-uniform mesh is and The relationship between them is: , is an integer.

[0117] Since the embodiments of the system part correspond to the embodiments of the method part, the embodiments of the system part refer to the description of the embodiments of the method part, which will not be described here. And it has the same beneficial effect as the fast three-dimensional forward modeling method with second-order calculation accuracy of the non-uniform grid model mentioned above.

[0118] Embodiment 3

[0119] This embodiment provides a computer-readable storage medium having a computer program stored thereon. When the program is executed by a processor, the steps in the fast three-dimensional forward modeling method with second-order calculation accuracy of the non-uniform grid model as described above are implemented.

[0120] Embodiment 4

[0121] This embodiment provides a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, the steps in the fast three-dimensional forward modeling method with second-order calculation accuracy of the non-uniform grid model as described above are implemented.

[0122] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A fast three-dimensional forward modeling method with second-order computational accuracy for non-uniform grid models, characterized by: include: Determine the non-uniform grid discretization method of the three-dimensional forward numerical model calculation area, and divide the three-dimensional forward numerical model calculation area; Based on the divided calculation area, the central difference method of non-adjacent field values ​​is used to calculate the partial derivatives of the magnetic field at the edges of the non-uniform grid, so that the electromagnetic field calculation results at this location have second-order calculation accuracy; Assemble the calculation equations of all discrete electric and magnetic fields in the calculation area to obtain the fully discrete equation of the electromagnetic field with second-order calculation accuracy; solve the equation to obtain the electric and magnetic field values ​​at the sampling points in the measurement space; The non-uniform grid discretization method of the calculation area of ​​the three-dimensional forward numerical model is: assuming that the model is x The direction is uniformly meshed, and the mesh size is ,exist y The direction is non-uniform grid division, and the non-uniform grid interface y = y The left side of 1 is the grid size The grid size is of the grid, and ; The central difference method of non-adjacent field values ​​is used to calculate the partial derivatives of the magnetic field at the edges of the non-uniform grid, so that the electromagnetic field calculation results at this location have second-order calculation accuracy, including: Determine the size and The positional relationship of the grid; Calculate the Taylor expansion of the discrete magnetic field at the center of the left grid surface at the non-uniform grid interface; based on and The relationship between them is used to calculate the Taylor expansion of the discrete magnetic field at the center of the non-adjacent grid surface on the right side of the non-uniform grid interface; The partial differential operator of the magnetic field at the grid edge is obtained by subtracting the Taylor expansion of the discrete magnetic field at the center of the left grid surface from the Taylor expansion of the discrete magnetic field at the center of the right non-adjacent grid surface of the non-uniform grid interface; The non-uniform mesh size that satisfies the second-order accuracy of the partial derivative calculation results of the magnetic field at the edges of the non-uniform mesh and The relationship between them is: , is an integer.

2. The fast three-dimensional forward modeling method with second-order calculation accuracy of non-uniform grid model according to claim 1, characterized in that: The partial derivative of the magnetic field at the edge of the non-uniform grid is: ， In the formula, is the partial derivative of the magnetic field at the edge of the non-uniform grid, represents the discrete magnetic field at the center of the left grid surface, represents the discrete magnetic field at the center of the right grid surface, Indicates that the magnetic field y The second derivative at 1, represents the higher-order error term with respect to the mesh size.

3. The fast three-dimensional forward modeling method with second-order calculation accuracy of non-uniform grid model according to claim 1, characterized in that: The fully discrete equation of the electromagnetic field with second-order calculation accuracy is: ， ， ， ， ， ， In the formula, i , j and k is the index number of the discrete electromagnetic field in space, where The index number representing the position of the non-uniform grid interface. and They are x and z Directional grid size, express The ratio of express x The virtual electric field in the direction of express y The virtual electric field in the direction of express z The virtual electric field in the direction of express n time x The electric field in the direction of express n time y The electric field in the direction of express n time z The electric field in the direction of express n time x Direction of the magnetic field, express n time y Direction of the magnetic field, express n time z Direction of the magnetic field, express n+ 1 moment x The electric field in the direction of express n+ 1 moment y The electric field in the direction of express n+ 1 moment z The electric field in the direction y The direction is non-uniform grid division, and the non-uniform grid interface y = y The left side of 1 is the grid size The grid size is The grid, , , , , and are the vacuum magnetic permeability and vacuum dielectric constant, respectively. σ is the conductivity, is the step time.

4. A fast three-dimensional forward modeling system with second-order computational accuracy for non-uniform grid models, characterized by: include: A calculation region partitioning module is used to determine a non-uniform grid discretization method for the calculation region of the three-dimensional forward numerical model and partition the calculation region of the three-dimensional forward numerical model; The second-order calculation accuracy module is used to calculate the partial derivatives of the magnetic field at the edges of the non-uniform grid based on the divided calculation area using the central difference method of non-adjacent field values, so that the electromagnetic field calculation results at this position have second-order calculation accuracy; The three-dimensional forward modeling module is used to assemble the calculation equations of all discrete electric and magnetic fields in the calculation area to obtain the fully discrete electromagnetic field equation with second-order calculation accuracy; the equation is solved to obtain the electric and magnetic field values ​​at the sampling points in the measurement space; In the calculation area partitioning module, the non-uniform grid discretization method of the calculation area of ​​the three-dimensional forward numerical model is: assuming that the model is x The direction is uniformly meshed, and the mesh size is ,exist y The direction is non-uniform grid division, and the non-uniform grid interface y = y The left side of 1 is the grid size The grid size is of the grid, and ; In the second-order calculation accuracy module, the central difference method of non-adjacent field values ​​is used to calculate the partial derivatives of the magnetic field at the edges of the non-uniform grid, so that the electromagnetic field calculation results at this location have second-order calculation accuracy, including: Determine the size and The positional relationship of the grid; Calculate the Taylor expansion of the discrete magnetic field at the center of the left grid surface at the non-uniform grid interface; based on and The relationship between them is used to calculate the Taylor expansion of the discrete magnetic field at the center of the non-adjacent grid surface on the right side of the non-uniform grid interface; The partial differential operator of the magnetic field at the grid edge is obtained by subtracting the Taylor expansion of the discrete magnetic field at the center of the left grid surface from the Taylor expansion of the discrete magnetic field at the center of the right non-adjacent grid surface of the non-uniform grid interface; In the second-order calculation accuracy module, the non-uniform mesh size that satisfies the second-order accuracy of the calculation results of the partial derivatives of the magnetic field at the edges of the non-uniform mesh and The relationship between them is: , is an integer.

Images